Lensed CMB power spectrum biases from masking extragalactic sources

The cosmic microwave background (CMB) is gravitationally lensed by large-scale structure, which distorts observations of the primordial anisotropies in any given direction. Averaged over the sky, this important effect is routinely modelled with via the lensed CMB power spectra. This accounts for the variance of this distortion, where the leading variance effect is quadratic in the lensing deflections. However, we show that if bright extragalactic sources correlated with the large-scale structure are masked in a CMB map, the power spectrum measured over the unmasked area using a standard pseudo-$C_\ell$ estimator has an additional linear lensing effect arising from correlations between the masked area and the lensing. This induces a scale-dependent average demagnification of the unlensed distance between unmasked pairs of observed points and a negative contribution to the CMB correlation function peaking at $\sim 10$ arcmin. We give simple analytic models for point sources and a threshold mask constructed on a correlated Gaussian foreground field. We demonstrate the consistency of their predictions for masks removing radio sources and peaks of Sunyaev-Zeldovich and cosmic infrared background emissions using realistic numerical simulations. We discuss simple diagnostics that can be used to test for the effect in the absence of a good model for the masked sources and show that by constructing specific masks the effect can be observed on Planck data. For masks employed in the analysis of Planck and other current data sets, the effect is likely to be negligible, but may become an important sub-percent correction for future surveys if substantial populations of resolved sources are masked.